mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, and theoretical

computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...

, tapering is a kind of

shape A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...

deformation. Just as an

affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...

, such as scaling or shearing, is a first-order model of shape deformation, tapering is a higher order deformation just as twisting and bending. Tapering can be thought of as non-constant scaling by a given tapering function. The resultant deformations can be

linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

or nonlinear. To create a nonlinear taper, instead of scaling in ''x'' and ''y'' for all ''z'' with constants as in: :

q= \begin
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & 1
\end p,

let ''a'' and ''b'' be functions of ''z'' so that: :

q= \begin
a(p_z) & 0 & 0 \\
0 & b(p_z) & 0 \\
0 & 0 & 1
\end p.

An example of a linear taper is

a(z) = \alpha_0 + \alpha_1z

, and a quadratic taper

a(z) = _0 + _1z + _2z^2

. As another example, if the parametric equation of a cube were given by ''ƒ''(''t'') = (''x''(''t''), ''y''(''t''), ''z''(''t'')), a nonlinear taper could be applied so that the cube's volume slowly decreases (or tapers) as the function moves in the positive ''z'' direction. For the given cube, an example of a nonlinear taper along ''z'' would be if, for instance, the function ''T''(''z'') = 1/(''a'' + ''bt'') were applied to the cube's equation such that ''ƒ''(''t'') = (''T''(''z'')''x''(''t''), ''T''(''z'')''y''(''t''), ''T''(''z'')''z''(''t'')), for some real constants ''a'' and ''b''.

References

{{reflist

External links

Computer Graphics Notes. University of Toronto. (See: Tapering).

3D Transformations. Brown University. (See: Nonlinear deformations).

ScienceWorld article on Tapering in Image Synthesis. Linear algebra Functions and mappings

See also

References

External links